6.2 properties of exponentsmarktoci.weebly.com/uploads/8/3/6/4/83643980/chapter6.2_1.pdf · section...

268 Chapter 6 Exponential Equations and Functions

Properties of Exponents6.2

How can you use inductive reasoning to

observe patterns and write general rules involving properties of exponents?

Work with a partner. Write the product of the two powers as a single power. Then, write a general rule for fi nding the product of two powers with the same base.

a. Sample: (34)(33) = (3 ⋅ 3 ⋅ 3 ⋅ 3)(3 ⋅ 3 ⋅ 3) = 37

b. (22)(23) =

c. (41)(45) =

d. (53)(55) = e. (x 2)(x 6) =

ACTIVITY: Writing a Rule for Products of Powers11

Work with a partner. Write the quotient of the two powers as a single power. Then, write a general rule for fi nding the quotient of two powers with the same base.

a. Sample: 34

— 32 =

3 ⋅ 3 ⋅ 3 ⋅ 3 —

3 ⋅ 3 = 32

b. 43

— 42 =

c.

25

— 22 =

d. x 6

— x 3

= e. 34

— 34 =

ACTIVITY: Writing a Rule for Quotients of Powers22

Work with a partner. Write the expression as a single power. Then, write a general rule for fi nding a power of a power.

a. Sample: (32)3 = (3 ⋅ 3)(3 ⋅ 3)(3 ⋅ 3) = 36

b. (22)4 =

c. (73)2 =

d. ( y 3)3 = e. (x 4)2 =

ACTIVITY: Writing a Rule for Powers of Powers33COMMON CORE

Exponents In this lesson, you will● simplify expressions

using the propertiesof exponents.

Learning StandardN.RN.2

Section 6.2 Properties of Exponents 269

6. IN YOUR OWN WORDS How can you use inductive reasoning to observe patterns and write general rules involving properties of exponents?

7. There are 33 small cubes in the cube below. Write an expression for the number of small cubes in the large cube at the right.

Use what you learned about exponents to complete Exercises 6 –11 on page 273.

Work with a partner. Write the expression as the product of two powers. Then, write a general rule for fi nding a power of a product.

a. Sample: (2 ⋅ 3)3 = (2 ⋅ 3)(2 ⋅ 3)(2 ⋅ 3) = (23)(33)

b. (2 ⋅ 5)2 =

c. (5 ⋅ 4)3 =

d. (6a )4 = e. (3x)2 =

ACTIVITY: Writing a Rule for Powers of Products44

Work with a partner. Write the expression as the quotient of two powers. Then, write a general rule for fi nding a power of a quotient.

a. Sample: ( 3 — 2

) 4 =

3 —

2 ⋅

3 —

2 ⋅

3 —

2 ⋅

3 —

2 =

3 ⋅ 3 ⋅ 3 ⋅ 3 —

2 ⋅ 2 ⋅ 2 ⋅ 2 =

34

— 24

b. ( 2 — 3

) 2 =

c. ( 4 —

3 )

3 =

d. ( x — 2

) 3 = e. ( a —

b )

4 =

ACTIVITY: Writing a Rule for Powers of Quotients55

View as ComponentsWhat are the different parts of the expressions? How does this help you rewrite the product?

Math Practice


Lesson6.2Lesson Tutorials

Product of Powers Property

Words To multiply powers with the same base, add their exponents.

Numbers 46 ⋅ 43 = 46 + 3 = 49 Algebra am ⋅ an = am + n

Quotient of Powers Property

Words To divide powers with the same base, subtract their exponents.

Numbers 46

— 43 = 46 − 3 = 43 Algebra

am

— an = am − n, where a ≠ 0

Power of a Power Property

Words To fi nd a power of a power, multiply the exponents.

Numbers (46)3 = 46 ⋅ 3 = 418 Algebra (am)n = amn

RememberFor any integer n and any nonzero integer a,

a0 = 1 and a− n = 1 —

an .

Exercises 12 –17

EXAMPLE Using Properties of Exponents11Simplify. Write your answer using only positive exponents.

a. 32 ⋅ 36 = 32 + 6 Product of Powers Property

= 38 Simplify.

b. (− 4)2

— (− 4)7 = (− 4)2 − 7 Quotient of Powers Property

= (− 4)− 5 Simplify.

= 1 —

(− 4)5 Defi nition of negative exponent

c. (z4)− 3 = z4 ⋅ (− 3) Power of a Power Property

= z− 12 Simplify.

= 1 —

z12 Defi nition of negative exponent

Simplify. Write your answer using only positive exponents.

1. 104 ⋅ 10− 6 2. x9 ⋅ x− 9 3. − 58

— − 54

4. y 6

— y7 5. (6− 2)− 5 6. (w12)5

The base is 3. Add the exponents.

The base is − 4. Subtract the exponents.

The base is z. Multiply the exponents.


Power of a Product Property

Words To fi nd a power of a product, fi nd the power of each factor and multiply.

Numbers (3 ⋅ 2)5 = 35 ⋅ 25 Algebra (ab)m = ambm

Power of a Quotient Property

Words To fi nd a power of a quotient, fi nd the power of the numerator and the power of the denominator and divide.

Numbers ( 3 — 2

) 5 =

35

— 25 Algebra ( a —

b )

m =

am

— bm , where b ≠ 0

Exercises 21 –26

EXAMPLE Using Properties of Exponents22Simplify. Write your answer using only positive exponents.

a. (− 1.5y)2 = (− 1.5)2 ⋅ y 2 Power of a Product Property

= 2.25y 2 Simplify.

b. ( a —

− 10 )

3 =

a3

— (− 10)3 Power of a Quotient Property

= − a3

— 1000

Simplify.

c. ( 2x —

3 )

− 5 =

(2x)− 5

— 3− 5 Power of a Quotient Property

= 35

— (2x)5 Defi nition of negative exponent

= 35

— 25x 5

Power of a Product Property

= 243

— 32x 5

Simplify.


7. (10y)− 3 8. ( − 4

— n

) 5

9. ( 1 —

2k 2 )

5 10. ( 6c

— 7

) −2


EXAMPLE Simplifying an Expression33Which expression represents the volume of the cylinder?

○A h2

— 2

○B πh2

— 4

○C πh3

— 2

○D πh3

— 4

V = πr 2h Formula for volume of a cylinder

= π ( h — 2

) 2(h) Substitute

h —

2 for r.

= π ( h2

— 22 ) (h) Power of a Quotient Property

= πh3

— 4

Simplify.

The correct answer is ○D .

h2

h

EXAMPLE Real-Life Application44

A jellyfi sh emits about 1.25 × 108 particles of light, or photons, in 6.25 × 10–4 second. How many photons does the jellyfi sh emit each second? Write your answer in scientifi c notation and in standard form.

Divide to fi nd the unit rate.

1.25 × 108

— 6.25 × 10− 4 Write the rate.

= 1.25

— 6.25

× 108

— 10− 4 Rewrite.

= 0.2 × 1012 Simplify.

= 2 × 1011 Write in scientifi c notation.

The jellyfi sh emits 2 × 1011, or 200,000,000,000 photons per second.

11. In Example 3, which expression represents the area of a base of the cylinder?

12. It takes the Sun about 2.3 × 108 years to orbit the center of the Milky Way. It takes Pluto about 2.5 × 102 years to orbit the Sun. How many times does Pluto orbit the Sun while the Sun completes one orbit around the Milky Way? Write your answer in scientifi c notation.

photons

seconds

RememberA number is written in scientifi c notation when it is of the form a × 10b, where 1 ≤ a < 10 and b is an integer.


Exercises6.2Help with Homework

9+(-6)=3

3+(-3)=

4+(-9)=

9+(-1)=

MATCHING Match the property with its example.

1. Quotient of Powers Property 2. Power of a Power Property

3. Power of a Quotient Property 4. Power of a Product Property

A. (45)2 = 45 ⋅

2 B. ( 5 — 2

) 4 =

54

— 24 C. (5 ⋅ 2)4 = 54 ⋅ 24 D.

45

— 42 = 45 − 2

5. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers.

Simplify 33 ⋅ 36. Simplify 33 + 6. Simplify 36 − 3. Simplify 36 ⋅ 33.

Simplify the expression.

6. (n 4)(n 3) 7. x 5

— x 3

8. (c5)3

9. (4b)3 10. ( k — 3

) 5 11.

(2a)6

— a2


12. 8− 2 ⋅ 87 13. b4 ⋅ b7 14. 127

— 122

15. d 5

— d 8

16. (55)4 17. (x 3)− 2

ERROR ANALYSIS Describe and correct the error in simplifying the expression.

18.

x 5 ⋅ x−2 = x 5 ⋅ (−2)

= x−10

= 1 —

x 10

✗ 19.

(m3)4 = m3 + 4

= m7✗

20. MICROSCOPE A microscope magnifi es an object 105 times. The length of an object is 102 nanometers. What is its magnifi ed length?

11



21. (6.2y)2 22. ( w — 4

) 4 23. ( −

6 —

d )

− 2

24. (7p)− 3 25. (− 5x)5 26. ( 3n3

— 4

) 2

27. ERROR ANALYSIS Describe and correct the error in simplifying the expression.

( x 3 —

3 ) 2 = (x 3)2

— 3

= x 6 —

3 ✗

28. OPEN-ENDED Use the properties of exponents to write three expressions equivalent to x 8.

29. REASONING Are the expressions (a4)2 and a42 equivalent? Explain

your reasoning.

30. GEOMETRY Consider Cube A and Cube B.

a. Which property of exponents should you use to fi nd the volume of each cube?

b. How can you use the Power of a Quotient Property to fi nd how many times greater the volume of Cube B is than the volume of Cube A?

31. SPHERE The volume V of a sphere is V = 4

— 3

πr 3,

where r is the radius. What is the volume of the sphere in terms of m and π ?

32. PROBABILITY The probability of rolling a 6 on a number cube is 1

— 6

.

The probability of rolling a 6 twice in a row is ( 1 — 6

) 2 =

1 —

36 .

a. Write an expression that represents the probability of rolling a 6 n times in a row.

b. What is the probability of rolling a 6 fi ve times in a row?

c. What is the probability of fl ipping heads on a coin fi ve times in a row?

22 33

2x

8x

A

B

2m2


Simplify the expression. (Section 6.1)

47. √—

48 48. √—

70

— 36

49. √—

180

— 121

50. MULTIPLE CHOICE Which of the following is the solution of x

— 3

< − 6? (Section 3.3)

○A x > − 2 ○B x < − 2 ○C x > − 18 ○D x < − 18

Evaluate the expression. Write your answer in scientifi c notation.

33. (3.4 × 102)(1.5 × 10− 5) 34. (6.1 × 10− 3)(8 × 109) 35. (4.8 × 10− 4)(7.2 × 10− 6)

36. (3 × 103)

— (4 × 105)

37. (6.4 × 10− 7)

— (1.6 × 10− 5)

38. (3.9 × 10− 5)

— (7.8 × 10− 8)


39. (6x 2y− 4)− 3 40. (2m)− 2n5

— − m4n− 3 41.

15b− 3c 4 —

(6b− 4c− 5)2

42. REASONING Write 8x 3y 3 as the power of a product.

43. COMPUTER CHIP The area of a rectangular computer chip is 112a 3b 2 square microns. The width is 8ab microns. What is the length?

44. PROBLEM SOLVING The speed of light is approximately 3 × 105 kilometers per second. The table shows the average distance each planet is from the Sun. How long does it take sunlight to reach Earth? Jupiter? Neptune?

45. RICHTER SCALE The Richter Scale is used to compare the intensities of earthquakes. An increase of 1 in magnitude on the Richter Scale represents a tenfold increase in intensity. An earthquake registers 7.4 on the Richter Scale and is followed by an aftershock that is 1000 times less intense. What is the magnitude of the aftershock?

46. PrecisionPrecision Find x and y when k 2x

— k y

= k13 and (k xk 2y )2 = k 28.

Explain how you found your answer.

44

PlanetAverage Distance from the Sun (km)

Mercury 5.8 × 107

Venus 1.1 × 108

Earth 1.5 × 108

Mars 2.3 × 108

Jupiter 7.8 × 108

Saturn 1.4 × 109

Uranus 2.9 × 109

Neptune 4.5 × 109

PlanetAverage Distance

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